MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The universal $R$-matrix of a quantized universal enveloping algebra is typically written as the product of two terms, one only involving elements of the Cartan, and one only involving elements of the upper and lower triangular pieces of $U_q(g)$.

How do I think about commuting these individual pieces past things like coproducts? Are calculations like this written somewhere?

share|cite|improve this question
up vote 3 down vote accepted

Another way to think about this, which I learned from Mark Haiman's class, is that there are actually $4$ natural coproducts on the quantum group. $\Delta\left(E\right)$ could be $E\otimes K + 1 \otimes E$, or you could move the $K$ to the other factor, or you could invert $K$, or you could do both. The individual parts of the $R$-matrix move between these four different coproducts. In particular, the quasi-$R$-matrix (iirc) turns the $K$ into a $K^{-1}$.

share|cite|improve this answer

The answer is in Lusztig's preposterously short PNAS paper on canonical bases for tensor products. He calls this element the "quasi-R-matrix."

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.